Hydrogen

Find the expectation value of for an electron in the ground state of Hydrogen. The normalized radial wave fu

If we use Schrodinger’s equation and the normalized radal wave function:

ψ(r,θ,Ф) = sqrt[(1/na)³(n-l-1)!/2n(n+1)!] e^-(ρ/2) ρ^l L(ρ)^(2l+1) Y(θ,Ф)

where:
ρ = 2r/na
a is the Bohr radius.
L(ρ)^(2l+1) are the Generalized Laguerre polynomials of degree n-l-1.
Y(θ,Ф) is a spherical harmonic

Then the energy levels of Hydrogen, including fine structure are given by:

E(n,j) = (-13.6 ev/n²) [1+(α²/n²) (n/(j+½)-¾) ]

where

α is the fine-structure constant (=1/137.036), and
j is an integer which is the angular momentum eigenvalue

Hence, the expectation value of an electron in the ground state of the Hydrogen atom is:

E(n=1, j=0) = (-13.6 ev) [1+(α²) (2-¾)] = (-13.6 ev) (1.00006)

Notice the answer is basically -13.6eV, which is the same as the result from the simple Bohr model, where:

E(n) = -me^4/8h²ε² (1/n²) = -13.6 (1/n²) eV

m – mass of the electron
q – charge of the electron
ε – the permittivity of free space

So E(ground state, n=1) = -13.6 eV

Lec 5 | MIT 3.091 Introduction to Solid State Chemistry

Atomic structure calculations: II. Hartree-Fock wavefunctions and radial expectation values : hydrogen to lawrencium …